Proof

Proof. The intended difference is $(1,z_n)\in \mathrm{int}{L}_m$. Events within either time slice are incomparable. For $n\ne k$, the spatial difference between $a_n$ and $b_k$ has norm at least $10|k-n|-1>1$, so those pairs are incomparable as well.

The closure of the cone generated by $(1,z_n)$ is $L_m$: the closure of their convex hull in the slice $t=1$ is the closed unit ball. Thus $\mathsf{Aff}$ holds with $K=L_m$, which satisfies $\mathsf{S}$.